A rational map $T$ of degree not less than two is known topreserve a measure, called the conformal measure, equivalent tothe Hausdorff measure of the same dimension as its Julia set $J$ andsupported there, with respect to which it is ergodic and evenexact. As a consequence of Birkhoff's pointwise ergodic theoremalmost every $z$ in $J$ with respect to the conformal measurehas an orbit that is asymptotically distributed on $J$ withrespect to this measure. As a counterpoint to this, the followingresult is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set$\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ ofdegree at least two we set\[S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}.\]We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$and $J$ are equal.